Model Answer
0 min readIntroduction
Project management often involves assessing the likelihood of completing a project within a specified timeframe. This assessment relies heavily on probabilistic techniques, recognizing that project durations are rarely fixed and are subject to uncertainty. Techniques like Program Evaluation and Review Technique (PERT) and Critical Path Method (CPM) are employed to estimate project completion probabilities. These methods utilize statistical analysis of individual task durations to determine the overall project duration and its associated probability. The question, "What is the probability of completing the project within 23 weeks?" necessitates a detailed understanding of these techniques and the data required for their application.
Understanding Project Completion Probability
Determining the probability of completing a project within a given timeframe requires a comprehensive understanding of the project's structure and the uncertainties associated with each task. The following steps are typically involved:
1. Project Network Diagram
The first step is to create a project network diagram, visually representing the project's tasks and their dependencies. This diagram identifies the critical path – the sequence of tasks that determines the shortest possible project duration.
2. Task Duration Estimation
Estimating the duration of each task is crucial. Instead of a single estimate, PERT uses three estimates:
- Optimistic Time (a): The shortest possible time to complete the task.
- Most Likely Time (m): The most realistic time to complete the task.
- Pessimistic Time (b): The longest possible time to complete the task.
The expected time (te) for each task is calculated using the formula: te = (a + 4m + b) / 6
3. Variance Calculation
The variance (σ2) for each task is calculated as: σ2 = ((b - a) / 6)2. This represents the uncertainty associated with the task duration.
4. Critical Path Analysis
Identifying the critical path is essential. Any delay in tasks along the critical path directly impacts the overall project completion time.
5. Project Duration Distribution
The Central Limit Theorem states that the distribution of the project completion time approaches a normal distribution as the number of independent tasks increases. Therefore, we can assume the project completion time follows a normal distribution.
6. Calculating Probability
To calculate the probability of completing the project within 23 weeks, we need to:
- Calculate the mean (μ) of the project completion time by summing the expected times (te) of all tasks on the critical path.
- Calculate the standard deviation (σ) of the project completion time by taking the square root of the sum of the variances (σ2) of all tasks on the critical path.
- Use the Z-score formula to determine the probability: Z = (X - μ) / σ, where X is the target completion time (23 weeks).
- Look up the Z-score in a standard normal distribution table to find the probability.
Example (Illustrative)
Let's assume a simplified critical path with two tasks:
| Task | Optimistic (a) | Most Likely (m) | Pessimistic (b) | Expected Time (te) | Variance (σ2) |
|---|---|---|---|---|---|
| Task A | 5 | 7 | 9 | 7.17 | 0.69 |
| Task B | 8 | 10 | 12 | 10.17 | 1.33 |
Total Expected Time (μ) = 7.17 + 10.17 = 17.34 weeks
Total Variance (σ2) = 0.69 + 1.33 = 2.02
Standard Deviation (σ) = √2.02 = 1.42 weeks
Z = (23 - 17.34) / 1.42 = 3.98
Looking up a Z-score of 3.98 in a standard normal distribution table gives a probability close to 1 (or 100%). This indicates a very high probability of completing the project within 23 weeks, given these hypothetical task durations.
Limitations
This analysis assumes task durations are independent. In reality, dependencies and resource constraints can introduce correlations, affecting the accuracy of the probability calculation. Furthermore, the normal distribution assumption may not always hold true, especially for projects with a small number of tasks.
Conclusion
In conclusion, calculating the probability of completing a project within a specific timeframe, like 23 weeks, requires a detailed project network, accurate task duration estimates (optimistic, most likely, and pessimistic), and statistical analysis using techniques like PERT/CPM. While the provided example illustrates the process, a real-world application necessitates comprehensive project data. Acknowledging the limitations of these methods and considering potential risks are crucial for effective project management and realistic probability assessments.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.